\(\int \frac {(c+d x^2)^2}{(a+b x^2)^{5/2} (e+f x^2)} \, dx\) [367]

Optimal result

Integrand size = 30, antiderivative size = 169 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {(b c-a d)^2 x}{3 a b (b e-a f) \left (a+b x^2\right )^{3/2}}+\frac {(b c-a d) \left (2 b^2 c e-a^2 d f+a b (4 d e-5 c f)\right ) x}{3 a^2 b (b e-a f)^2 \sqrt {a+b x^2}}+\frac {(d e-c f)^2 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} (b e-a f)^{5/2}} \] Output:

1/3*(-a*d+b*c)^2*x/a/b/(-a*f+b*e)/(b*x^2+a)^(3/2)+1/3*(-a*d+b*c)*(2*b^2*c* 
e-a^2*d*f+a*b*(-5*c*f+4*d*e))*x/a^2/b/(-a*f+b*e)^2/(b*x^2+a)^(1/2)+(-c*f+d 
*e)^2*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^(1/2)/(-a*f+b* 
e)^(5/2)
 

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {(b c-a d) x \left (2 b^2 c e x^2+a b \left (3 c e+4 d e x^2-5 c f x^2\right )+a^2 \left (3 d e-6 c f-d f x^2\right )\right )}{3 a^2 (b e-a f)^2 \left (a+b x^2\right )^{3/2}}-\frac {(d e-c f)^2 \arctan \left (\frac {-f x \sqrt {a+b x^2}+\sqrt {b} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-b e+a f}}\right )}{\sqrt {e} (-b e+a f)^{5/2}} \] Input:

Integrate[(c + d*x^2)^2/((a + b*x^2)^(5/2)*(e + f*x^2)),x]
 

Output:

((b*c - a*d)*x*(2*b^2*c*e*x^2 + a*b*(3*c*e + 4*d*e*x^2 - 5*c*f*x^2) + a^2* 
(3*d*e - 6*c*f - d*f*x^2)))/(3*a^2*(b*e - a*f)^2*(a + b*x^2)^(3/2)) - ((d* 
e - c*f)^2*ArcTan[(-(f*x*Sqrt[a + b*x^2]) + Sqrt[b]*(e + f*x^2))/(Sqrt[e]* 
Sqrt[-(b*e) + a*f])])/(Sqrt[e]*(-(b*e) + a*f)^(5/2))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.49, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {419, 25, 398, 224, 219, 291, 221, 401, 25, 27, 298, 224, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 419

\(\displaystyle -\frac {\int -\frac {\left (d x^2+c\right ) \left (d f a^2-2 b c f a+b^2 (d e-c f) x^2+b^2 c e\right )}{\left (b x^2+a\right )^{5/2}}dx}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (d x^2+c\right ) \left (d f a^2-2 b c f a+b^2 (d e-c f) x^2+b^2 c e\right )}{\left (b x^2+a\right )^{5/2}}dx}{(b e-a f)^2}-\frac {f (d e-c f) \int \frac {d x^2+c}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {\int \frac {\left (d x^2+c\right ) \left (d f a^2-2 b c f a+b^2 (d e-c f) x^2+b^2 c e\right )}{\left (b x^2+a\right )^{5/2}}dx}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\int \frac {\left (d x^2+c\right ) \left (d f a^2-2 b c f a+b^2 (d e-c f) x^2+b^2 c e\right )}{\left (b x^2+a\right )^{5/2}}dx}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\int \frac {\left (d x^2+c\right ) \left (d f a^2-2 b c f a+b^2 (d e-c f) x^2+b^2 c e\right )}{\left (b x^2+a\right )^{5/2}}dx}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\int \frac {\left (d x^2+c\right ) \left (d f a^2-2 b c f a+b^2 (d e-c f) x^2+b^2 c e\right )}{\left (b x^2+a\right )^{5/2}}dx}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\int \frac {\left (d x^2+c\right ) \left (d f a^2-2 b c f a+b^2 (d e-c f) x^2+b^2 c e\right )}{\left (b x^2+a\right )^{5/2}}dx}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {\frac {x \left (c+d x^2\right ) (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}-\frac {\int -\frac {b \left (3 a b d (d e-c f) x^2+c \left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right )\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a b}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {b \left (3 a b d (d e-c f) x^2+c \left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right )\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a b}+\frac {x \left (c+d x^2\right ) (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {3 a b d (d e-c f) x^2+c \left (2 d f a^2+b (d e-5 c f) a+2 b^2 c e\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x \left (c+d x^2\right ) (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {\frac {3 a d (d e-c f) \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {x (b c-a d) (-5 a c f+3 a d e+2 b c e)}{a \sqrt {a+b x^2}}}{3 a}+\frac {x \left (c+d x^2\right ) (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {3 a d (d e-c f) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {x (b c-a d) (-5 a c f+3 a d e+2 b c e)}{a \sqrt {a+b x^2}}}{3 a}+\frac {x \left (c+d x^2\right ) (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{(b e-a f)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {3 a d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (d e-c f)}{\sqrt {b}}+\frac {x (b c-a d) (-5 a c f+3 a d e+2 b c e)}{a \sqrt {a+b x^2}}}{3 a}+\frac {x \left (c+d x^2\right ) (b c-a d) (b e-a f)}{3 a \left (a+b x^2\right )^{3/2}}}{(b e-a f)^2}-\frac {f (d e-c f) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} f}-\frac {(d e-c f) \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} f \sqrt {b e-a f}}\right )}{(b e-a f)^2}\)

Input:

Int[(c + d*x^2)^2/((a + b*x^2)^(5/2)*(e + f*x^2)),x]
 

Output:

(((b*c - a*d)*(b*e - a*f)*x*(c + d*x^2))/(3*a*(a + b*x^2)^(3/2)) + (((b*c 
- a*d)*(2*b*c*e + 3*a*d*e - 5*a*c*f)*x)/(a*Sqrt[a + b*x^2]) + (3*a*d*(d*e 
- c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b])/(3*a))/(b*e - a*f)^2 
 - (f*(d*e - c*f)*((d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*f) - 
((d*e - c*f)*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt 
[e]*f*Sqrt[b*e - a*f])))/(b*e - a*f)^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 401

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 419

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[b*((b*e - a*f)/(b*c - a*d)^2)   Int[(c + d*x^2)^( 
q + 2)*((e + f*x^2)^(r - 1)/(a + b*x^2)), x], x] - Simp[1/(b*c - a*d)^2   I 
nt[(c + d*x^2)^q*(e + f*x^2)^(r - 1)*(2*b*c*d*e - a*d^2*e - b*c^2*f + d^2*( 
b*e - a*f)*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && Gt 
Q[r, 1]
 

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(\frac {-a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (c f -d e \right )^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )+2 \left (a d -b c \right ) \left (\left (-\frac {1}{2} d e +\frac {1}{6} d f \,x^{2}+c f \right ) a^{2}-\frac {\left (\frac {\left (-5 c f +4 d e \right ) x^{2}}{3}+c e \right ) b a}{2}-\frac {e \,b^{2} c \,x^{2}}{3}\right ) \sqrt {\left (a f -b e \right ) e}\, x}{\left (a f -b e \right )^{2} \sqrt {\left (a f -b e \right ) e}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\)	\(163\)
default	\(\text {Expression too large to display}\)	\(1558\)

Input:

int((d*x^2+c)^2/(b*x^2+a)^(5/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

2*(-1/2*a^2*(b*x^2+a)^(3/2)*(c*f-d*e)^2*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b 
*e)*e)^(1/2))+(a*d-b*c)*((-1/2*d*e+1/6*d*f*x^2+c*f)*a^2-1/2*(1/3*(-5*c*f+4 
*d*e)*x^2+c*e)*b*a-1/3*e*b^2*c*x^2)*((a*f-b*e)*e)^(1/2)*x)/((a*f-b*e)*e)^( 
1/2)/(b*x^2+a)^(3/2)/(a*f-b*e)^2/a^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (151) = 302\).

Time = 2.75 (sec) , antiderivative size = 1154, normalized size of antiderivative = 6.83 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(5/2)/(f*x^2+e),x, algorithm="fricas")
 

Output:

[1/12*(3*(a^4*d^2*e^2 - 2*a^4*c*d*e*f + a^4*c^2*f^2 + (a^2*b^2*d^2*e^2 - 2 
*a^2*b^2*c*d*e*f + a^2*b^2*c^2*f^2)*x^4 + 2*(a^3*b*d^2*e^2 - 2*a^3*b*c*d*e 
*f + a^3*b*c^2*f^2)*x^2)*sqrt(b*e^2 - a*e*f)*log(((8*b^2*e^2 - 8*a*b*e*f + 
 a^2*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f) 
*x^3 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + 
e^2)) + 4*((2*(b^4*c^2 + a*b^3*c*d - 2*a^2*b^2*d^2)*e^3 - (7*a*b^3*c^2 - 2 
*a^2*b^2*c*d - 5*a^3*b*d^2)*e^2*f + (5*a^2*b^2*c^2 - 4*a^3*b*c*d - a^4*d^2 
)*e*f^2)*x^3 + 3*((a*b^3*c^2 - a^3*b*d^2)*e^3 - (3*a^2*b^2*c^2 - 2*a^3*b*c 
*d - a^4*d^2)*e^2*f + 2*(a^3*b*c^2 - a^4*c*d)*e*f^2)*x)*sqrt(b*x^2 + a))/( 
a^4*b^3*e^4 - 3*a^5*b^2*e^3*f + 3*a^6*b*e^2*f^2 - a^7*e*f^3 + (a^2*b^5*e^4 
 - 3*a^3*b^4*e^3*f + 3*a^4*b^3*e^2*f^2 - a^5*b^2*e*f^3)*x^4 + 2*(a^3*b^4*e 
^4 - 3*a^4*b^3*e^3*f + 3*a^5*b^2*e^2*f^2 - a^6*b*e*f^3)*x^2), -1/6*(3*(a^4 
*d^2*e^2 - 2*a^4*c*d*e*f + a^4*c^2*f^2 + (a^2*b^2*d^2*e^2 - 2*a^2*b^2*c*d* 
e*f + a^2*b^2*c^2*f^2)*x^4 + 2*(a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^ 
2*f^2)*x^2)*sqrt(-b*e^2 + a*e*f)*arctan(1/2*sqrt(-b*e^2 + a*e*f)*((2*b*e - 
 a*f)*x^2 + a*e)*sqrt(b*x^2 + a)/((b^2*e^2 - a*b*e*f)*x^3 + (a*b*e^2 - a^2 
*e*f)*x)) - 2*((2*(b^4*c^2 + a*b^3*c*d - 2*a^2*b^2*d^2)*e^3 - (7*a*b^3*c^2 
 - 2*a^2*b^2*c*d - 5*a^3*b*d^2)*e^2*f + (5*a^2*b^2*c^2 - 4*a^3*b*c*d - a^4 
*d^2)*e*f^2)*x^3 + 3*((a*b^3*c^2 - a^3*b*d^2)*e^3 - (3*a^2*b^2*c^2 - 2*a^3 
*b*c*d - a^4*d^2)*e^2*f + 2*(a^3*b*c^2 - a^4*c*d)*e*f^2)*x)*sqrt(b*x^2 ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+c)**2/(b*x**2+a)**(5/2)/(f*x**2+e),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(5/2)/(f*x^2+e),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 768 vs. \(2 (151) = 302\).

Time = 0.15 (sec) , antiderivative size = 768, normalized size of antiderivative = 4.54 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {{\left (\frac {{\left (2 \, b^{6} c^{2} e^{3} + 2 \, a b^{5} c d e^{3} - 4 \, a^{2} b^{4} d^{2} e^{3} - 9 \, a b^{5} c^{2} e^{2} f + 9 \, a^{3} b^{3} d^{2} e^{2} f + 12 \, a^{2} b^{4} c^{2} e f^{2} - 6 \, a^{3} b^{3} c d e f^{2} - 6 \, a^{4} b^{2} d^{2} e f^{2} - 5 \, a^{3} b^{3} c^{2} f^{3} + 4 \, a^{4} b^{2} c d f^{3} + a^{5} b d^{2} f^{3}\right )} x^{2}}{a^{2} b^{5} e^{4} - 4 \, a^{3} b^{4} e^{3} f + 6 \, a^{4} b^{3} e^{2} f^{2} - 4 \, a^{5} b^{2} e f^{3} + a^{6} b f^{4}} + \frac {3 \, {\left (a b^{5} c^{2} e^{3} - a^{3} b^{3} d^{2} e^{3} - 4 \, a^{2} b^{4} c^{2} e^{2} f + 2 \, a^{3} b^{3} c d e^{2} f + 2 \, a^{4} b^{2} d^{2} e^{2} f + 5 \, a^{3} b^{3} c^{2} e f^{2} - 4 \, a^{4} b^{2} c d e f^{2} - a^{5} b d^{2} e f^{2} - 2 \, a^{4} b^{2} c^{2} f^{3} + 2 \, a^{5} b c d f^{3}\right )}}{a^{2} b^{5} e^{4} - 4 \, a^{3} b^{4} e^{3} f + 6 \, a^{4} b^{3} e^{2} f^{2} - 4 \, a^{5} b^{2} e f^{3} + a^{6} b f^{4}}\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} + \frac {{\left (b d^{4} e^{4} - 4 \, b c d^{3} e^{3} f + 6 \, b c^{2} d^{2} e^{2} f^{2} - 4 \, b c^{3} d e f^{3} + b c^{4} f^{4}\right )} \arctan \left (-\frac {2 \, b^{\frac {3}{2}} d^{2} e^{3} - 4 \, b^{\frac {3}{2}} c d e^{2} f - a \sqrt {b} d^{2} e^{2} f + 2 \, b^{\frac {3}{2}} c^{2} e f^{2} + 2 \, a \sqrt {b} c d e f^{2} - a \sqrt {b} c^{2} f^{3} + {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} \sqrt {b} d^{2} e^{2} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} \sqrt {b} c d e f + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} \sqrt {b} c^{2} f^{2}\right )} f}{2 \, {\left (\sqrt {-b e^{2} + a e f} b d^{2} e^{2} - 2 \, \sqrt {-b e^{2} + a e f} b c d e f + \sqrt {-b e^{2} + a e f} b c^{2} f^{2}\right )}}\right )}{{\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} {\left (b^{2} e^{2} - 2 \, a b e f + a^{2} f^{2}\right )} \sqrt {-b e^{2} + a e f}} \] Input:

integrate((d*x^2+c)^2/(b*x^2+a)^(5/2)/(f*x^2+e),x, algorithm="giac")
 

Output:

1/3*((2*b^6*c^2*e^3 + 2*a*b^5*c*d*e^3 - 4*a^2*b^4*d^2*e^3 - 9*a*b^5*c^2*e^ 
2*f + 9*a^3*b^3*d^2*e^2*f + 12*a^2*b^4*c^2*e*f^2 - 6*a^3*b^3*c*d*e*f^2 - 6 
*a^4*b^2*d^2*e*f^2 - 5*a^3*b^3*c^2*f^3 + 4*a^4*b^2*c*d*f^3 + a^5*b*d^2*f^3 
)*x^2/(a^2*b^5*e^4 - 4*a^3*b^4*e^3*f + 6*a^4*b^3*e^2*f^2 - 4*a^5*b^2*e*f^3 
 + a^6*b*f^4) + 3*(a*b^5*c^2*e^3 - a^3*b^3*d^2*e^3 - 4*a^2*b^4*c^2*e^2*f + 
 2*a^3*b^3*c*d*e^2*f + 2*a^4*b^2*d^2*e^2*f + 5*a^3*b^3*c^2*e*f^2 - 4*a^4*b 
^2*c*d*e*f^2 - a^5*b*d^2*e*f^2 - 2*a^4*b^2*c^2*f^3 + 2*a^5*b*c*d*f^3)/(a^2 
*b^5*e^4 - 4*a^3*b^4*e^3*f + 6*a^4*b^3*e^2*f^2 - 4*a^5*b^2*e*f^3 + a^6*b*f 
^4))*x/(b*x^2 + a)^(3/2) + (b*d^4*e^4 - 4*b*c*d^3*e^3*f + 6*b*c^2*d^2*e^2* 
f^2 - 4*b*c^3*d*e*f^3 + b*c^4*f^4)*arctan(-1/2*(2*b^(3/2)*d^2*e^3 - 4*b^(3 
/2)*c*d*e^2*f - a*sqrt(b)*d^2*e^2*f + 2*b^(3/2)*c^2*e*f^2 + 2*a*sqrt(b)*c* 
d*e*f^2 - a*sqrt(b)*c^2*f^3 + ((sqrt(b)*x - sqrt(b*x^2 + a))^2*sqrt(b)*d^2 
*e^2 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*sqrt(b)*c*d*e*f + (sqrt(b)*x - sq 
rt(b*x^2 + a))^2*sqrt(b)*c^2*f^2)*f)/(sqrt(-b*e^2 + a*e*f)*b*d^2*e^2 - 2*s 
qrt(-b*e^2 + a*e*f)*b*c*d*e*f + sqrt(-b*e^2 + a*e*f)*b*c^2*f^2))/((b*d^2*e 
^2 - 2*b*c*d*e*f + b*c^2*f^2)*(b^2*e^2 - 2*a*b*e*f + a^2*f^2)*sqrt(-b*e^2 
+ a*e*f))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^{5/2}\,\left (f\,x^2+e\right )} \,d x \] Input:

int((c + d*x^2)^2/((a + b*x^2)^(5/2)*(e + f*x^2)),x)
 

Output:

int((c + d*x^2)^2/((a + b*x^2)^(5/2)*(e + f*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 2122, normalized size of antiderivative = 12.56 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

int((d*x^2+c)^2/(b*x^2+a)^(5/2)/(f*x^2+e),x)
 

Output:

( - 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x 
**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*b**2*c**2*f**2 + 6*sqrt( 
e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt 
(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*b**2*c*d*e*f - 3*sqrt(e)*sqrt(a*f - 
 b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x 
)/(sqrt(e)*sqrt(b)))*a**4*b**2*d**2*e**2 - 6*sqrt(e)*sqrt(a*f - b*e)*atan( 
(sqrt(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)* 
sqrt(b)))*a**3*b**3*c**2*f**2*x**2 + 12*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt 
(a*f - b*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt( 
b)))*a**3*b**3*c*d*e*f*x**2 - 6*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b 
*e) - sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a** 
3*b**3*d**2*e**2*x**2 - 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - 
sqrt(f)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b**4 
*c**2*f**2*x**4 + 6*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f 
)*sqrt(a + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b**4*c*d*e 
*f*x**4 - 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) - sqrt(f)*sqrt(a 
 + b*x**2) - sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**2*b**4*d**2*e**2*x** 
4 - 3*sqrt(e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x 
**2) + sqrt(f)*sqrt(b)*x)/(sqrt(e)*sqrt(b)))*a**4*b**2*c**2*f**2 + 6*sqrt( 
e)*sqrt(a*f - b*e)*atan((sqrt(a*f - b*e) + sqrt(f)*sqrt(a + b*x**2) + s...